When women daven (pray), even by ourselves, it is convention to daven with male-gendered words ("modeh" vs "moda" as one of many examples). Why is this; are there sources that say we should be doing otherwise? If so, when did this tradition get started?
The original siddur did not include a version for women. Changes to the format for women began later on. R. Jacob Emden (in his siddur commentary) suggested emending the morning blessings for women, but didn't recommend it. Chid"a (Avodat haKodesh 2 5 22) allowed the changes, along with Eshel Avraham (the Buchacher, OC 46 4) and Rivvos Efraim (1 37 2) quoting R. Chaim Kanievsky. Many siddurim have adopted this approach.
Opposing views include Siach Tefilla (10) Lubavicher Rebbe (Halacha uMinhag 1 50), citing the need to adhere to established custom, which never altered the service for women. Furthermore, it is well-known that the male gender is also the neutral gender in Biblical Hebrew, thus it includes both men and women.
We do find precedent for changing the service for women in Grace after Meals, see Rema (OC 187 3). But in that case, it is more than just a grammar issue.
More on this topic in the discussion at this link (Hebrew).
Most of our prayers are worded in the plural ("we" are asking for something) or in the second person (from You, G-d - and YKVK is a masculine word, that's why it's atah, not at.)
However, the few times where there would be a grammatical difference for women; according to the publishers of Rinat Yisrael, women should say the blessings / prayers according to the rules of proper grammar.
First off, there are many Siddurim that bring the differences. Secondly, the Hida tweaks the Tefila a little to fix this problem as well.
Can't find the Hida, but I know that Yalkut Yosef (1:1:1:3) says to say "moda ani."
An Italian siddur written in 1471 includes in the dawn blessings, "she'asitani isha velo ish", 'that You made me a woman and not a man' (as well as the changes "shifha" for "eved" and "nokhrit" for "goy", as mentioned by others). So, I don't think we should be quick to say that using language applicable to women was a late change.